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The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity

Publisher:

Springer

Number of Pages:

324

Price:

79.95

ISBN:

9781441978370

Special relativity in Minkowski spacetime is sometimes regarded as mere stepping-stone to the mathematically rich study of general relativity. Yet, as the current book shows, it has beauty and depth all its own. This text brings sophisticated mathematical structures and tools to play, yet much of the work would be accessible to a motivated undergraduate. Although it addresses all the usual topics of special relativity (and more), this is by no means a standard introduction to special relativity.

The author lays out his goal very clearly: “It is the intention of this monograph to provide an introduction to the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics.” He then proceeds to accomplish this admirably. Along the way he is very careful to make clear distinctions between doing mathematics and appealing to physical arguments to interpret the mathematics.

The “event world” of special relativity is captured in a model that includes Minkowski spacetime (a four-dimension real vector space with a nondegenerate symmetric bilinear form of index one) and an associated group of orthogonal transformations — the Lorentz group. The book’s long first chapter explores the geometry of the model and establishes some of the basic results like time dilation, causality conditions and Lorentz contraction. One mathematical highlight is a theorem due to Zeeman: every causal automorphism is a composition of a translation, a dilation, and an orthochronous orthogonal transformation. Another is Penrose’s theorem on the apparent shape of a sphere moving at relativistic speeds. (It’s probably not what you’d first think. The key is that the relevant group of transformations is isomorphic to the group of linear fractional transformations of the Riemann sphere.)

The second chapter introduces charged particles, the classical Lorentz force law and the electromagnetic field into Minkowski space. The author characterizes the electromagnetic field at a point in Minkowski spacetime as a skew symmetric transformation that tells a charged particle with a given velocity what change in its momentum will occur when it is subject to that field. In this chapter, as throughout, we get a lot of mileage from linear algebra. By the end we get to the (source-free) Maxwell equations and some of their solutions in a Coulomb field and with a uniformly moving charged particle.

The author then introduces the algebraic theory of spinors in Minkowski spacetime in the third chapter. His intention in part is to recast in spinor form much of the work on the structure of the electromagnetic field from the previous chapter. Since he believes the best way to approach spinors is via group representations, it also gives him an opportunity to introduce students to another important tool of mathematical physics.

The final chapter, which is new to this second edition, gently explores the steps that are needed to move past special relativity and introduce gravitation into spacetime. By providing some fairly basic background in manifolds, Riemannian and Lorentzian metrics, geodesics and curvature, the reader is led first to Einstein’s field equations and then to a consideration of the deSitter solution of those equations.

To those accustomed to reading mathematical physics, this book may seem unnecessarily fussy. But even those who are comfortable with less rigorous arguments would have to admit that the careful handling here avoids many of the confusions that turn up in more casual treatments. Also, the underlying mathematics is wonderful, worth studying for its own sake.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.